On The Collatz Conjecture

Yesterday, during the monthly Florida Bibliophile Society meeting, a book collecting former math teacher with a profound interest in Lewis Carroll (of Alice & Wonderland fame) made some interesting comments on an area of mathematics I find quite intriguing. He spoke in particular on the Collatz Conjecture. A conjecture, in mathematical terms, is a supposition that appears to be always true but is not rigorously proven. There are, of course, several ways to prove a statement, which would change the conjecture into a theorem (which is rigorously proven and is followed by three of my favorite latin words--quod erat demonstradum, which mean "it has been demonstrated/proven."

Logic gives us several ways of proving a statement. One is a proof by contradiction. In this proof, we assume that the theorem is not true and then demonstrate that this leads to a contradiction with what is known to be true. For example, if we assumed a theorem was not true and then showed that if this was the case then 1 = 0, then we would prove by contradiction that the theorem was true. This is a particularly elegant form of proof, and I quite enjoy that as well. Other proofs seek to demonstrate that a statement boils down, eventually, to already proven statements, so that the theorem rests on the shoulders of prior statements (standing on the shoulder of giants, as it were). The immediate implications of a given theorem are considered corollaries, and are added to the pool of proven statements as well. Alternatively, theorems may be proven exhaustively, such that within a given range one can demonstrate that the theorem is true because one has exhausted all of the possibilities. I dislike this manner of proof because I tend to get rather bored of calculating algorithms and my attention has the tendency to wander, leading to mistakes in arithmetic. Since this is apparently a common affliction, generally computers are used in this form beyond the simplest of cases, because their attention does not wander (unless they use a Microsoft operating system and have a blue screen error, but that is another subject entirely).

I particularly enjoy conjectures because they are often composed of rather simple statements and can be verified to very large numbers, but because they resist proof by contradiction or traditional methods of proof, and because they make such grand claims that they cannot be proven exhaustively. One such example of this sort of conjecture is that the digits of pi and Euler's constant do not have a repetitive pattern, making them transcendental numbers. Well, this has not been proven (or disproven, for that matter) despite the fact that computers have calculated pi and Euler's constant to extreme lengths and have found (in the case of pi at least) that there are roughly equal numbers of each digit despite no discernable pattern.

Collatz's conjecture is likewise appealing to me because the conjecture sets two rather simple rules, one for odd numbers and another for even numbers, and these rules have been demonstrated to be correct to very large numbers, but because there are an infinite number of postive numbers, the conjecture cannot be proven true exhaustively. Furthermore, the conjecture has resisted a proof by contradiction or the traditional deductive proof.

The Collatz Conjecture has two rules:
If a number is odd, multiply by three and add one (3x +1),
If a number is even, divide the number in half (x/2).
The conjecture states that, eventually, any number will reduce, given these two rules, to one.

Some examples:

1 4 2 1
2 1
3 10 5 16 8 4 2 1
4 2 1
5 16 8 4 2 1
6 3 10 5 16 8 4 2 1
7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
8 4 2 1
9 28 14 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
10 5 16 8 4 2 1

And so on...

It becomes apparent rather quickly that there is some kind of pattern in the way the rule works, and that once a sequence reaches a number already present in an existing sequence then it follows the same trail (for example, 11 would be 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1). However, the proof of this conjecture has evaded the attempts of mathematicians of far greater ability than I (though, to be honest, it is not a difficult task to find mathematicians of greater ability than I am).

I consider such examples as this simple conjecture as evidence that the world in which we live, even when we look only at such narrow areas as mathematics which appear at first capable of exhaustive rational proof, nonetheless possesses aspects which we hold to be true even though we cannot demonstrate them to be true by logic. The world we live in, that is, requires something we call faith. If this is true in the world of number theory, it is certainly even more true in those aspects of life that are not as quantifiable. This may not make some mathematicians very happy, but it should nonetheless provide food for thought for those of us who are able to handle a bit more uncertainty and are able to reflect upon intelligence greater than our own.

Three Vignettes On The Life Of A Graduate Student

My life as a graduate student provides me with numerous opportunities for dry humor and (sometimes) dispassionate reflection on the quirks of higher education and my own place within it. Knowing that graduate education is still somewhat rare in the United States, and particularly rare in my family (I am the first person in my family to go to graduate school since my great-great uncle Willis David Matthias got his Th.D from Princeton Theological Seminary in 1927, but that is another story), hopefully my musings are not too esoteric or obscure.

On Americans In Graduate School

As an American in a graduate engineering program, I am a rather unusual fellow (regardless of whether I would be unusual on account of my own inherent quirkiness). The vast majority of my fellow graduate students in the School of Engineering, even at the University of South Florida, are not American citizens. Those that are Americans are often older students (many of whom have children only a little younger than myself), and many of the other students come from such diverse lands as China, India, South Korea, Turkey, France, Columbia, Venezuela, Saudi Arabia, and Morocco. I happen to enjoy studying with such people, as they provide me the opportunity to learn about other lands and backgrounds beyond my own (and I tend to seek out such knowledge anyway). So, for me it is not the presence of people from other lands that is noteworthy. What is noteworthy is the absence of other Americans, particularly young Americans, from my classes. I know I am an odd fellow, but being interested in graduate school in engineering should not make me that unusual, I would think. What makes this rather amusing is that immigration law, which for me personally is a rather non-issue, becomes a very important matter for many of my classmates. One classmate, an undergraduate in a combination graduate/undergraduate program at USF, is from Columbia, and she was unable to go to a recent academic conference she helped organize because it was in Cancun and she did not have travel visa privileges. Other friends of mine face difficulties in finding employment because of a shortage of H1-B visas, with the liklihood that they will have to return to their home nations.

This becomes a problem in that many professions (such as civil engineering, my own) are moving towards making professional credentials more difficult to obtain, often to the point of requiring graduate school. With the above-inflation rise in college costs and the increasing length of time of non-productive (in economic terms) years spent acquiring an education rather than earning a living, coupled with the shortage in young people willing or able to study advanced degrees in science and engineering (and the difficulties in legally hiring educated foreigners to these positions) we face some serious problems. There appear to be no shortage of ambitious young adults in MBA programs, but when it comes to more technical studies the shortage is most glaring (and I have seen it with my own eyes). It appears as if my generation of students has understood that we value the acquisition of money above all else--above integrity, above truth/knowledge--and they have chosen their academic careers accordingly. What are we to do to ensure our nation has enough intellectuals to fill those emptying ivory towers, or even the ranks of middle-class professions like scientists and engineers?

A Nexus Of Cash And Corruption

It is a common thread of mine to comment on corruption in this world wherever it can be found, and recently I have found another occasion to comment on this. A scandal is breaking in the world of academic financing, as numerous university officials have been found to have had overly cozy financial relationships with a clique of financial organizations engaged in predatory lending to university students. While the universities implicated so far in the scandal are on the east coast (Penn, St. John's, and so forth), there are plenty of schools in other parts of the United States that engage in the same practices (and I know of some schools on the west coast that they could look into as well).

The university officials implicated in this scandal (and some government officials as well) received lucrative benefits from a few preferred lenders who apparently gave the universities (and those officials) bonuses based on how many students chose particular loan packages that contained terms advantageous to the lending companies and not as advantageous to the students themselves. Andrew Cuomo, the attorney general for the state of New York (a position that in recent years has inspired rather fiercely anti-corruption officials), apparently began the investigation at the behest of a lender who was apparently shut out of elite college lending on account of being unable to schmooze with the university officials and being unwilling/financially unable to engage in the graft/corruption that such college lending apparently entails. Perhaps the AG of New York should go after the credit card companies that camp out on university sidewalks continually with free t-shirts to give out. Maybe they are paying off university officials as well. That would not be surprising.

A Parking Rant

Actually, one of the benefits of being a commuter student (as I am now) is avoiding such people. When I lived on campus as an undergraduate on the left coast, I could only avoid such people at food speed, and those who know me should realize my foot speed is not terribly fast. Unfortunately, one of the annoyances of being a commuter student is the problem of finding adequate parking.

A few days ago, I received a survey from the University of South Florida about parking, which gave me the opportunity to rant to the people responsible for the travesty that is engineering school parking. (It is a rare treat in my life to be able to rant directly, and anonymously, to people who make my life more difficult than I would prefer). I was bemused by the attempts of the survey to advocate the use of Tampa's rather unacceptable public transportation service (known as HARTLine), as well as advocate future light rail projects to connect USF to the airport and a few other major places. (The question of how someone is supposed to get from where they live to the light rail and back, much less go to work as well, is left unstated).

I took the opportunity of the survey to make a few choice (but non-profane) comments about the incompetence of those responsible for parking at USF. Of course, these are the same geniuses who removed an entire parking lot from student use (for who knows what reason) some weeks ago while the Spring Semester is still in session. Parking was crowded enough before, now the situation is even more untenable. I do not appreciate having to walk half a mile from my car to class, especially not when I have to pass half-empty lots I cannot use because they are reserved for staff. I take such things rather personally, and as a sign of the lack of genuine interest people in charge have about doing what is best for those they purportedly serve. It is distressing that it is so hard to find leaders who are not corrupt and incompetent. If leaders were as competent in serving as they were in advocating their own pet agendas and in avoiding due regulation and accountability, I would have much less to rant about. I think that would be a good thing all the way around. For now, I study what is around me, and occasionally report on it.

Why Is The Beam Always In Our Eye?

One of the most pointed verses in the Bible concerning our behavior to other people is Matthew 7:3: "Why do you look at the speck that is in your brother's eye, but do not notice the log that is in your own eye? (NASB)" One question I have long had about this verse is, why is the beam (or log) always in our eye, and why is the speck in the eye of the other person? Today I would like to look at at least a few possible answers.

On Specks

The answer to the question of why the speck is the eye of the other person seems a bit easier to answer, so I will tackle that one first. What we see of other people is rather limited. Even a fairly obvious person (I would probably qualify as one of these) still keeps a fair amount of information inside simply because not everything can be communicated to other people without a large amount of time spent with someone in a wide variety of different situations. Furthermore, much of what we do communicate verbally and nonverbally is subject to interpretation (and misinterpretation), which presents a further barrier to understanding other people. Therefore, even for people whose living is spent critiquing (that is how I pay my bills, after all) only have a small amount of material available to judge someone on.

This need not be a bad thing. We are not to be judgmental after all (much easier said than done), so it would be salutory for those of us who do tend to be a tad more hypercritical than we probably should be to recognize that we may be wrong (sometimes really wrong) and mean it. However, even when we judge something correctly in other people, we must realize that even here our information may be incomplete. No mercy will be shown to those who show no mercy, and when we do not know why someone acts the way they do, we may err in being harsh to those who suffer and struggle greatly. Consequently, we may be too easy on those who are better at hiding their sins (or are politically powerful enough that most people with a modicum of sense quail at challenging). So, what we see is often imperfect and that we see imperfectly. Therefore we cannot but see specks in the eyes of others, as their logs are often quite hidden to us.

On Logs

On the other hand, we all have logs in our eyes (I have mine, and I am blessed to have other people around me who seem to take great pleasure in reminding me about them, often). We need not be too hard on ourselves--God knows our situations better than we do--but we do need to look realistically at ourselves and understand that we all fall far short of our goal. Without the pardon given to us through the sacrifice of Jesus Christ, salvation would be far beyond our reach, as if we were trying to stand on the ground and grasp for the stars in the night sky.

There are many reasons we have logs. For one, most people do not examine themselves particularly closely. We selectively compare ourselves to others and justify ourselves by looking at how other people fall short and how we have been wronged (I say this knowing I do the same thing, often, myself). In doing so, even when we see the worse parts of our own nature, we mitigate them and try to explain away our own problems through forces beyond our control while denying this privilege to others, what is called the fundamental attribution error. Furthermore, by not dealing with our own problems, we have a skewed perception of reality, which prevents us from seeing our world clearly and makes us less able to correctly judge other people.

What To Do Then?

How do we avoid this difficulty? Besides remaining in a state of blissfully ignorant hypocrisy, there are really two solutions. One is not to judge at all, and the other is to learn how to judge wisely. Our society right now seems to point towards not judging at all (though here it is only those people who wish to live righteously who are judged, in a perversion of the concept of judging). Not judging basically amounts to this: you do what you want to do, I'll do what I want to do, and as long as you don't hurt or judge me I won't judge you. This is incorrect, because if we desire a just society that is suitable for human existence, we must realize that no act is without consequences for other people, no matter how private it may seem. I could go at length about this, but a common example should suffice. For example, many people think that buying cocaine is a decision that effects only the drug user. This is false. The purchase of drugs is but one link in a rather complicated picture that involves a lot of people. Besides harming the user himself (itself a sin, as we are to honor God's temple--our bodies), the act of buying drugs supports a whole legion of people who make their livings in illegal ways (drug dealers, laboratories, etc.) who often combine drugs with other criminal behavior such as prostitution, theft, violence. These people, in turn, depend on the sources of their drugs from international drug traffickers who themselves are involved in crimes such as illegal immigration, terrorism, the international slave trade, massacres of populations in drug growing areas, and so forth. When we act, we can either be a part of a nexus of good or a nexus of evil. We cannot pretend, though, that what we do has no consequences outside of ourselves.

That leaves us with the choice of judging wisely. In doing so, we need to be aware (at least as much as possible) about what is right and wrong, and about why what is simple for us may be difficult for someone else (and the reverse is true also). In seeking to understand others, we may find that they have much to teach us and we have much to learn, even when we set out to teach them about their errors. Also, we must separate the sin from the sinner. We are all sinners, all worthy of the death penalty for our actions, and all (potentially) covered by the grace of God should we choose to accept the offer. We all have much to struggle with in this life, and we should recognize it as such. If we spend the limited time and energy we have on this earth honestly and openly seeking to live rightly (and increase in knowledge so that we have a better idea of what it is we are to do and how it is we are to do it), then we will speak louder about sin than we would through a million jeremiads against the sins of others. Sometimes we speak loudest through our own example--do you know what your example is saying? That is a frightening picture to consider indeed.